Group synchronization of coupled harmonic oscillators without velocity measurements

This paper investigates the group synchronization of coupled harmonic oscillators over a directed network topology in the absence of velocity measurements. Each harmonic oscillator can only obtain the sampled position states relative to its neighbors at a series of discrete-time instants. Two distributed control protocols are proposed based on the impulsive control and sampled-data control strategies. Theoretical analysis shows that the desired sampling period is determined by the position gain and the eigenvalues of the Laplacian matrix associated with the network topology. Some necessary and sufficient conditions for group synchronization are analytically established in virtue of matrix analysis, graph theory and polynomial Schur stability theory. Different to the synchronization criteria presented in the form of linear matrix inequality or general inequality, which may need to be verified, this paper explicitly gives the ranges for all feasible sampling periods. A significant feature of the synchronization criteria is that certain functional relationships between the feasible sampling period, the largest real part of the eigenvalues of the Laplacian matrix, the largest ratio of the imaginary part to the real part of the eigenvalues of the Laplacian matrix (if there exist complex eigenvalues) and the position gain are analytically established. Some effective iterative methods are then derived to calculate the endpoints of the feasible range of the sampling periods for achieving group synchronization. Finally, numerical experiments further verify the correctness of the theoretical results.